Theorem icocncflimc 37761

Description: Limit at the lower bound, of a continuous function defined on a left closed right open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
icocncflimc.a	|- ( ph -> A e. RR )
icocncflimc.b	|- ( ph -> B e. RR* )
icocncflimc.altb	|- ( ph -> A < B )
icocncflimc.f	|- ( ph -> F e. ( ( A [,) B ) -cn-> CC ) )

Assertion

Ref	Expression
icocncflimc	|- ( ph -> ( F ` A ) e. ( ( F |` ( A (,) B ) ) lim CC A ) )

Proof of Theorem icocncflimc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icocncflimc.f	. . 3 |- ( ph -> F e. ( ( A [,) B ) -cn-> CC ) )
2		icocncflimc.a	. . . . 5 |- ( ph -> A e. RR )
3	2	rexrd 9687	. . . 4 |- ( ph -> A e. RR* )
4		icocncflimc.b	. . . 4 |- ( ph -> B e. RR* )
5	2	leidd 10177	. . . 4 |- ( ph -> A <_ A )
6		icocncflimc.altb	. . . 4 |- ( ph -> A < B )
7	3, 4, 3, 5, 6	elicod 11682	. . 3 |- ( ph -> A e. ( A [,) B ) )
8	1, 7	cnlimci 22837	. 2 |- ( ph -> ( F ` A ) e. ( F lim CC A ) )
9		cncfrss 21916	. . . . . . . 8 |- ( F e. ( ( A [,) B ) -cn-> CC ) -> ( A [,) B ) C_ CC )
10	1, 9	syl 17	. . . . . . 7 |- ( ph -> ( A [,) B ) C_ CC )
11		ssid 3450	. . . . . . 7 |- CC C_ CC
12		eqid 2450	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
13		eqid 2450	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) )
14		eqid 2450	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( ( TopOpen ` ℂfld ) ↾t CC )
15	12, 13, 14	cncfcn 21934	. . . . . . 7 |- ( ( ( A [,) B ) C_ CC /\ CC C_ CC ) -> ( ( A [,) B ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
16	10, 11, 15	sylancl 667	. . . . . 6 |- ( ph -> ( ( A [,) B ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
17	1, 16	eleqtrd 2530	. . . . 5 |- ( ph -> F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
18	12	cnfldtopon 21796	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
19	18	a1i 11	. . . . . . 7 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
20		resttopon 20170	. . . . . . 7 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( A [,) B ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) e. (TopOn ` ( A [,) B ) ) )
21	19, 10, 20	syl2anc 666	. . . . . 6 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) e. (TopOn ` ( A [,) B ) ) )
22	12	cnfldtop 21797	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Top
23		unicntop 37365	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
24	23	restid 15325	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
25	22, 24	ax-mp 5	. . . . . . 7 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
26	25	cnfldtopon 21796	. . . . . 6 |- ( ( TopOpen ` ℂfld ) ↾t CC ) e. (TopOn ` CC )
27		cncnp 20289	. . . . . 6 |- ( ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) e. (TopOn ` ( A [,) B ) ) /\ ( ( TopOpen ` ℂfld ) ↾t CC ) e. (TopOn ` CC ) ) -> ( F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) <-> ( F : ( A [,) B ) --> CC /\ A. x e. ( A [,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) CnP ( ( TopOpen ` ℂfld ) ↾t CC ) ) ` x ) ) ) )
28	21, 26, 27	sylancl 667	. . . . 5 |- ( ph -> ( F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) <-> ( F : ( A [,) B ) --> CC /\ A. x e. ( A [,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) CnP ( ( TopOpen ` ℂfld ) ↾t CC ) ) ` x ) ) ) )
29	17, 28	mpbid 214	. . . 4 |- ( ph -> ( F : ( A [,) B ) --> CC /\ A. x e. ( A [,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) CnP ( ( TopOpen ` ℂfld ) ↾t CC ) ) ` x ) ) )
30	29	simpld 461	. . 3 |- ( ph -> F : ( A [,) B ) --> CC )
31		ioossico 11720	. . . 4 |- ( A (,) B ) C_ ( A [,) B )
32	31	a1i 11	. . 3 |- ( ph -> ( A (,) B ) C_ ( A [,) B ) )
33		eqid 2450	. . 3 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,) B ) u. { A } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,) B ) u. { A } ) )
34	2	recnd 9666	. . . . . . 7 |- ( ph -> A e. CC )
35	23	ntrtop 20079	. . . . . . . . 9 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = CC )
36	22, 35	ax-mp 5	. . . . . . . 8 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = CC
37		undif 3847	. . . . . . . . . . 11 |- ( ( A [,) B ) C_ CC <-> ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) = CC )
38	10, 37	sylib 200	. . . . . . . . . 10 |- ( ph -> ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) = CC )
39	38	eqcomd 2456	. . . . . . . . 9 |- ( ph -> CC = ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) )
40	39	fveq2d 5867	. . . . . . . 8 |- ( ph -> ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = ( ( int ` ( TopOpen ` ℂfld ) ) ` ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) ) )
41	36, 40	syl5eqr 2498	. . . . . . 7 |- ( ph -> CC = ( ( int ` ( TopOpen ` ℂfld ) ) ` ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) ) )
42	34, 41	eleqtrd 2530	. . . . . 6 |- ( ph -> A e. ( ( int ` ( TopOpen ` ℂfld ) ) ` ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) ) )
43	42, 7	elind 3617	. . . . 5 |- ( ph -> A e. ( ( ( int ` ( TopOpen ` ℂfld ) ) ` ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) ) i^i ( A [,) B ) ) )
44	22	a1i 11	. . . . . 6 |- ( ph -> ( TopOpen ` ℂfld ) e. Top )
45		ssid 3450	. . . . . . 7 |- ( A [,) B ) C_ ( A [,) B )
46	45	a1i 11	. . . . . 6 |- ( ph -> ( A [,) B ) C_ ( A [,) B ) )
47	23, 13	restntr 20191	. . . . . 6 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( A [,) B ) C_ CC /\ ( A [,) B ) C_ ( A [,) B ) ) -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) ) ` ( A [,) B ) ) = ( ( ( int ` ( TopOpen ` ℂfld ) ) ` ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) ) i^i ( A [,) B ) ) )
48	44, 10, 46, 47	syl3anc 1267	. . . . 5 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) ) ` ( A [,) B ) ) = ( ( ( int ` ( TopOpen ` ℂfld ) ) ` ( ( A [,) B ) u. ( CC \ ( A [,) B ) ) ) ) i^i ( A [,) B ) ) )
49	43, 48	eleqtrrd 2531	. . . 4 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) ) ` ( A [,) B ) ) )
50	7	snssd 4116	. . . . . . . . 9 |- ( ph -> { A } C_ ( A [,) B ) )
51		ssequn2 3606	. . . . . . . . 9 |- ( { A } C_ ( A [,) B ) <-> ( ( A [,) B ) u. { A } ) = ( A [,) B ) )
52	50, 51	sylib 200	. . . . . . . 8 |- ( ph -> ( ( A [,) B ) u. { A } ) = ( A [,) B ) )
53	52	eqcomd 2456	. . . . . . 7 |- ( ph -> ( A [,) B ) = ( ( A [,) B ) u. { A } ) )
54	53	oveq2d 6304	. . . . . 6 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,) B ) u. { A } ) ) )
55	54	fveq2d 5867	. . . . 5 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,) B ) u. { A } ) ) ) )
56		snunioo1 37607	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { A } ) = ( A [,) B ) )
57	3, 4, 6, 56	syl3anc 1267	. . . . . 6 |- ( ph -> ( ( A (,) B ) u. { A } ) = ( A [,) B ) )
58	57	eqcomd 2456	. . . . 5 |- ( ph -> ( A [,) B ) = ( ( A (,) B ) u. { A } ) )
59	55, 58	fveq12d 5869	. . . 4 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,) B ) ) ) ` ( A [,) B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,) B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
60	49, 59	eleqtrd 2530	. . 3 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,) B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
61	30, 32, 10, 12, 33, 60	limcres 22834	. 2 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC A ) = ( F lim CC A ) )
62	8, 61	eleqtrrd 2531	1 |- ( ph -> ( F ` A ) e. ( ( F |` ( A (,) B ) ) lim CC A ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   A.wral 2736   \ cdif 3400   u. cun 3401   i^i cin 3402   C_ wss 3403   {csn 3967   class class class wbr 4401   |` cres 4835   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   RR*cxr 9671   < clt 9672   (,)cioo 11632   [,)cico 11634   ↾_t crest 15312   TopOpenctopn 15313  ℂ_fldccnfld 18963   Topctop 19910  TopOnctopon 19911   intcnt 20025   Cn ccn 20233   CnP ccnp 20234   -cn->ccncf 21901   lim CC climc 22810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-ntr 20028  df-cn 20236  df-cnp 20237  df-xms 21328  df-ms 21329  df-cncf 21903  df-limc 22814

This theorem is referenced by:  fourierdlem46  38010